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On Nonlinear Forbidden 01 Matrices: A Refutation of a FürediHajnal Conjecture
"... A matrix A P t0, 1u m n is said to avoid a forbidden pattern P P t0, 1u k l if no k l submatrix of A matches P, where a 0 in P can match either a 0 or 1 in A. Let ExpP, nq be the maximum weight (i.e., number of 1s) of an n n matrix avoiding the pattern P or all patterns in the set P. The theory of f ..."
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A matrix A P t0, 1u m n is said to avoid a forbidden pattern P P t0, 1u k l if no k l submatrix of A matches P, where a 0 in P can match either a 0 or 1 in A. Let ExpP, nq be the maximum weight (i.e., number of 1s) of an n n matrix avoiding the pattern P or all patterns in the set P. The theory of forbidden submatrices subsumes many extremal problems in combinatorics and graph theory, including DavenportSchinzel sequences and their generalizations, Stanley and Wilf’s permutation avoidance problem, and Turántype subgraph avoidance problems. Forbidden submatrices have found many applications in discrete geometry and the analysis of both geometric and nongeometric algorithms. In general terms, to bound the complexity of an arrangement of objections or the running time of an algorithm, one transcribes the objects or operations as a 01 matrix that provably avoids some forbidden pattern or collection of patterns P. This method is useful only to the extent that ExpP, nq can be tightly bounded, for specific P s or whole classes of P s. A 01 matrix can be interpreted as the incidence matrix of a bipartite graph where vertices on either side of the bipartition are ordered. In 1992, Füredi and Hajnal conjectured that imposing
Origins of nonlinearity in DavenportSchinzel sequences
, 2009
"... A generalized DavenportSchinzel sequence is one over a finite alphabet that excludes subsequences isomorphic to a fixed forbidden subsequence. The fundamental problem in this area is bounding the maximum length of such sequences. Following Klazar, we let Expσ, nq be the maximum length of a sequence ..."
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A generalized DavenportSchinzel sequence is one over a finite alphabet that excludes subsequences isomorphic to a fixed forbidden subsequence. The fundamental problem in this area is bounding the maximum length of such sequences. Following Klazar, we let Expσ, nq be the maximum length of a sequence over an alphabet of size n excluding subsequences isomorphic to σ. It has been proved that for every σ, Expσ, nq is either linear or very close to linear. In particular it is Opn2 αpnqOp1q q, where α is the inverseAckermann function and Op1q depends on σ. In much the same way that the complete graphs K5 and K3,3 represent the minimal causes of nonplanarity, there must exist a set ΦNonlin of minimal nonlinear forbidden subsequences. Very little is known about the size or membership of ΦNonlin. In this paper we construct an infinite antichain of nonlinear forbidden subsequences which, we argue, strongly supports the conjecture that ΦNonlin is itself infinite. Perhaps the most novel contribution of this paper is a succinct, humanly readable code for expressing the structure of forbidden subsequences.
Degrees of Nonlinearity in Forbidden 01 Matrix Problems
"... A 01 matrix A is said to avoid a forbidden 01 matrix (or pattern) P if no submatrix of A matches P, where a 0 in P matches either 0 or 1 in A. The theory of forbidden matrices subsumes many extremal problems in combinatorics and graph theory such as bounding the length of DavenportSchinzel sequen ..."
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Cited by 3 (2 self)
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A 01 matrix A is said to avoid a forbidden 01 matrix (or pattern) P if no submatrix of A matches P, where a 0 in P matches either 0 or 1 in A. The theory of forbidden matrices subsumes many extremal problems in combinatorics and graph theory such as bounding the length of DavenportSchinzel sequences and their generalizations, Stanley and Wilf’s permutation avoidance problem, and Turántype subgraph avoidance problems. In addition, forbidden matrix theory has proved to be a powerful tool in discrete geometry and the analysis of both geometric and nongeometric algorithms. Clearly a 01 matrix can be interpreted as the incidence matrix of a bipartite graph in which vertices on each side of the partition are ordered. Füredi and Hajnal conjectured that if P corresponds to an acyclic graph then the maximum weight (number of 1s) in an n × n matrix avoiding P is O(n log n). Our first result is a refutation of this conjecture. We exhibiting n × n matrices with weight Θ(n log n log log n) that avoid a relatively small acyclic matrix. The matrices are constructed via two complementary composition operations for 01 matrices. Our second result is a simplified proof that there is an infinite antichain (with respect
On the Structure and Composition of Forbidden Sequences, with Geometric Applications
, 2010
"... Forbidden substructure theorems have proved to be among of the most versatile tools in bounding the complexity of geometric objects and the running time of geometric algorithms. To apply them one typically transcribes an algorithm execution or geometric object as a sequence over some alphabet or a 0 ..."
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Forbidden substructure theorems have proved to be among of the most versatile tools in bounding the complexity of geometric objects and the running time of geometric algorithms. To apply them one typically transcribes an algorithm execution or geometric object as a sequence over some alphabet or a 01 matrix, proves that this object avoids some subsequence or submatrix σ, then uses an off the shelf bound on the maximum size of such a σfree object. As a historical trend, expanding our library of forbidden substructure theorems has led to better bounds and simpler analyses of the complexity of geometric objects. We establish new and tight bounds on the maximum length of generalized DavenportSchinzel sequences, which are those whose subsequences are not isomorphic to some fixed sequence σ. (The standard DavenportSchinzel sequences restrict σ to be of the form abab · · ·.) 1. We prove that Nshaped forbidden subsequences (of the form abc · · · xyzyx · · · cbabc · · · xyz) have a linear extremal function. Our proof dramatically improves an earlier one of Klazar and Valtr in the leading constants and overall simplicity. This result tightens the (astronomical) leading constants in Valtr’s O(n log n) bound on geometric graphs without
Tight bounds on the maximum size of a set of permutations with bounded VCdimension
 In Proc. Symposium on Discrete Algorithms
, 2012
"... The VCdimension of a family P of npermutations is the largest integer k such that the set of restrictions of the permutations in P on some ktuple of positions is the set of all k! permutation patterns. Let rk(n) be the maximum size of a set of npermutations with VCdimension k. Raz showed that r2 ..."
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The VCdimension of a family P of npermutations is the largest integer k such that the set of restrictions of the permutations in P on some ktuple of positions is the set of all k! permutation patterns. Let rk(n) be the maximum size of a set of npermutations with VCdimension k. Raz showed that r2(n) grows exponentially in n. We show that r3(n) = 2 Θ(nlogα(n)) and for every t ≥ 1, we have r2t+2(n) = 2 Θ(nα(n)t) and r2t+3(n) = 2 O(nα(n)t logα(n)) We also study the maximum number pk(n) of 1entries in an n × n (0,1)matrix with no (k + 1)tuple of columns containing all (k+1)permutation matrices. We determine that p3(n) = Θ(nα(n)) and p2t+2(n) = n2 (1/t!)α(n)t ±O(α(n) t−1) for every t ≥ 1. We also show that for every positive s there is a slowly growing function ζs(m) (for example ζs(m) = 2 O(α(s−3)/2 (m)) for every odd s ≥ 5) satisfying the following. For all positive integers m,n,B and every m×n (0,1)matrix M with ζs(m)Bn 1entries, the rows of M can be partitioned into s intervals so that some ⌊Bn/m⌋tuple of columns contains at least B 1entries in each of the intervals. 1